On approximation theorems for controllability of non-linear parabolic problems

Kumar, Anil ; Joshi, Mohan C. ; Pani, Amiya K. (2007) On approximation theorems for controllability of non-linear parabolic problems IMA Journal of Mathematical Control and Information, 24 (1). pp. 115-136. ISSN 0265-0754

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Official URL: http://imamci.oxfordjournals.org/content/24/1/115....

Related URL: http://dx.doi.org/10.1093/imamci/dnl012


In this paper, we consider the following control system governed by the non-linear parabolic differential equation of the form: ∂y(/t)/∂t + Ay(t)=f(t,y(t)) + u(t), t ε [0,T], y(0) =y0 where A is a linear operator with dense domain and f(t, y) is a non-linear function. We have proved that under Lipschitz continuity assumption on the non-linear function f(t, y), the set of admissible controls is non-empty. The optimal pair (u, y) is then obtained as the limit of the optimal pair sequence {(un, yn)}, where un is a minimizer of the unconstrained problem involving a penalty function arising from the controllability constraint and yn is the solution of the parabolic non-linear system defined above. Subsequently, we give approximation theorems which guarantee the convergence of the numerical schemes to optimal pair sequence. We also present numerical experiment which shows the applicability of our result.

Item Type:Article
Source:Copyright of this article belongs to Oxford University Press.
Keywords:Controllability; Optimal Control; Non-linear Parabolic System; Penalty Function; Nemytskii Operator; CO-semigroup; Lipschitz Continuity; Generalized Hammerstein Equation
ID Code:81950
Deposited On:09 Feb 2012 04:47
Last Modified:09 Feb 2012 04:47

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